Divisibility

Gauss famously said that mathematics is the queen of sciences and number theory is the queen of mathematics. Divisibility is the first step on the path to understanding number theory.

So of course the AMC tests how well you know it.

Examples

You probably know that $14$ is divisible by $7$ , and that the prime factorization of $30$ is $2 * 3 * 5$ .

You probably also know that any integer that ends in a $5$ or $0$ is divisible by $5$ , and that any integer ending in an even digit is itself even.

But do you also know the simple rules that tell you whether a number is divisible by $3, 4, 6, 8, 9,$ and/or $11$ ?
And do you know how to tell how many factors a number has in all, given its prime factorization?
And do you know an expression for the product of all of a number’s factors?

The quizzes for this section cover all this and more.